Khovanov homology and Heegaard Floer homology are powerful invariants for knots and three-manifolds, which were discovered around the year 2000, and which have since stirred a tremendous amount of research activity. In particular, both Khovanov homology and Heegaard Floer homology have been used to give new proofs of the topological Milnor conjecture and of the existence of exotic smooth structures on the open four-ball. Previously, such results had only been accessible via gauge theory. While Khovanov homology is defined combinatorially, via a construction which is motivated by the representation theory of quantum groups, Heegaard Floer homology is defined analytically, through moduli spaces of solutions of differential equations. Around 2008, Heegaard Floer homology was extended to an invariant for three-manifolds with non-empty boundary, called "bordered Floer homology", which can be used to compute Heegaard Floer homology combinatorially whenever a decomposition of a three-manifold into suitable smaller pieces is given. A main goal of this project is to study the relationship between bordered Floer homology and Khovanov homology for tangles. Comparing these two theories will expectedly shed more light on the geometric content of Khovanov homology and thus make Khovanov homology more suited to applications. Moreover, the envisioned relationship between bordered Floer homology and Khovanov homology will provide an example of a perhaps more general connection between symplectic geometry and representation theory. Other goals of this project are to develop new homology theories for contact three-manifolds, and to analyze the properties of Khovanov homology groups of n-cables of knots.
Mathematicians have long been interested in classifying topological spaces that are locally three-dimensional (like our physical universe) or locally four-dimensional (like four-dimensional space-time). Related to the problem of classifying such spaces is the problem of classifying knotted loops embedded in a given three-dimensional space. Over the past two decades, mathematicians have used ideas coming from several different areas of mathematics and mathematical physics (in particular from symplectic geometry, quantum field theory, string theory, and loop quantum gravity) to develop powerful new tools for classifying knots and low-dimensional spaces. The most notable ones among these new tools are Khovanov homology and Heegaard Floer homology. This proposal aims to investigate the connections between certain generalizations of Khovanov homology and Heegaard Floer homology, and to use these connections to study knot theoretical problems. Mathematical knot theory has applications in biomedical research, where it is used to study the processes that are responsible for unraveling DNA strands during cell division. Thus, this project is important not only from a theoretical perspective, but also for its potential applications to biomedical sciences.
